Visualize the solid. It is like an inverted cone. But the cone angle is not a constant. The outer periphery of the solid (base of the cone like solid) is given by x² + y² = 2². or, r² = 2²
x= r cosФ and y = r sinФ
dA in a z plane : (dr) (r dФ)
The limits are 0<= r <= 2 and 0<= Ф <= π/2 for the first octant.
dV = z dA = (x + y) dr r dФ = r² (cosФ + sinФ) dr dФ
so the volume is 16/3